3.6.35 \(\int \frac {(a+b \tanh ^{-1}(c x)) (d+e \log (f+g x^2))}{x} \, dx\) [535]

Optimal. Leaf size=93 \[ a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )-\frac {1}{2} b d \text {PolyLog}(2,-c x)+\frac {1}{2} b d \text {PolyLog}(2,c x)+\frac {1}{2} a e \text {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+b e \text {Int}\left (\frac {\tanh ^{-1}(c x) \log \left (f+g x^2\right )}{x},x\right ) \]

[Out]

b*e*CannotIntegrate(arctanh(c*x)*ln(g*x^2+f)/x,x)+a*d*ln(x)+1/2*a*e*ln(-g*x^2/f)*ln(g*x^2+f)-1/2*b*d*polylog(2
,-c*x)+1/2*b*d*polylog(2,c*x)+1/2*a*e*polylog(2,1+g*x^2/f)

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Rubi [A]
time = 0.18, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x,x]

[Out]

a*d*Log[x] + (a*e*Log[-((g*x^2)/f)]*Log[f + g*x^2])/2 - (b*d*PolyLog[2, -(c*x)])/2 + (b*d*PolyLog[2, c*x])/2 +
(a*e*PolyLog[2, 1 + (g*x^2)/f])/2 + b*e*Defer[Int][(ArcTanh[c*x]*Log[f + g*x^2])/x, x]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx &=d \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx+e \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{x} \, dx\\ &=a d \log (x)-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)+(a e) \int \frac {\log \left (f+g x^2\right )}{x} \, dx+(b e) \int \frac {\tanh ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx\\ &=a d \log (x)-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)+\frac {1}{2} (a e) \text {Subst}\left (\int \frac {\log (f+g x)}{x} \, dx,x,x^2\right )+(b e) \int \frac {\tanh ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx\\ &=a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)+(b e) \int \frac {\tanh ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx-\frac {1}{2} (a e g) \text {Subst}\left (\int \frac {\log \left (-\frac {g x}{f}\right )}{f+g x} \, dx,x,x^2\right )\\ &=a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)+\frac {1}{2} a e \text {Li}_2\left (1+\frac {g x^2}{f}\right )+(b e) \int \frac {\tanh ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x,x]

[Out]

Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x, x]

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Maple [A]
time = 2.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x))*(d+e*ln(g*x^2+f))/x,x)

[Out]

int((a+b*arctanh(c*x))*(d+e*ln(g*x^2+f))/x,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))*(d+e*log(g*x^2+f))/x,x, algorithm="maxima")

[Out]

a*d*log(x) + integrate(1/2*b*(log(c*x + 1) - log(-c*x + 1))*e*log(g*x^2 + f)/x + 1/2*b*d*(log(c*x + 1) - log(-
c*x + 1))/x + a*e*log(g*x^2 + f)/x, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))*(d+e*log(g*x^2+f))/x,x, algorithm="fricas")

[Out]

integral((b*d*arctanh(c*x) + a*d + (b*arctanh(c*x)*e + a*e)*log(g*x^2 + f))/x, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right ) \left (d + e \log {\left (f + g x^{2} \right )}\right )}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x))*(d+e*ln(g*x**2+f))/x,x)

[Out]

Integral((a + b*atanh(c*x))*(d + e*log(f + g*x**2))/x, x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))*(d+e*log(g*x^2+f))/x,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x) + a)*(e*log(g*x^2 + f) + d)/x, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atanh(c*x))*(d + e*log(f + g*x^2)))/x,x)

[Out]

int(((a + b*atanh(c*x))*(d + e*log(f + g*x^2)))/x, x)

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